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0 votes
0 answers
63 views

Feller semigroups and fractional operators

Have Feller semigroups been used to investigate the properties of the Cauchy problem associated with the fractional Laplacian (just like they have been used to study local degenerate second order ...
2 votes
1 answer
4k views

What “mild solution” means, and how to find it?

In this paper: Existence and uniqueness of a classical solution to a functional-differential abstract nonlocal Cauchy problem Byszewski studied this form of functional-differential nonlocal problem (1)...
3 votes
1 answer
507 views

Chain rules for Dini Derivative

Could someone provides some references for the chain rule concerning Dini derivatives. For example, let $f(\cdot) \in \mathcal{C}^1\left( \mathbb{R} ; \mathbb{R}\right)$, and $g(\cdot) \in \mathcal{C}\...
-3 votes
1 answer
392 views

A generalization of Chebyshev's sum inequality

From some my previous questions here and here and well-known rearrangement inequality. I pose an inequality as follows and I am looking for the proof or a reference. Inequality: Let $y=f(x,y)$ is ...
4 votes
1 answer
860 views

Lebesgue's integrability condition in several variables

The well known Lebesgue's condition of Riemann integrability says that a bounded function in one variable $f\colon [a,b] \to \mathbb{R}$ is Riemann integrable if and only if it is continuous almost ...
5 votes
1 answer
795 views

How to define transfinite derivatives of a function?

There are all manners of theories generalizing the notion of derivative. Amongst them is the fractional calculus, a rich theory which gives a sense to the derivation and integration of non-integer (i....
4 votes
1 answer
203 views

If $T_1T_2 = T_2T_1$, why $r(T_1 + T_2) \leq r(T_1) + r(T_2)$?

Let $T_1$ and $T_2$ be two bounded linear operators in a complex banach space $X$. If $T_1T_2 = T_2T_1$, I want to know how to show that $$ r(T_1+T_2) \leq r(T_1) + r(T_2), $$ where $r(A)$ ...
1 vote
0 answers
50 views

Comparison of (square) of a function and its Fourier transform in an integral

I am completely stuck on a comparison between $f(t)^2$ and $\hat{f}(t)^2$ in an integral. Considering $f(t)$ of rapid decrease at infinity such that near zero: $f(t) \sim_0 t^{-\frac{1}{2}- \alpha}+o(...
8 votes
0 answers
110 views

Connected component optimization

For an open set $A\subset[0,1]^d$, denote the connected components of $A$ by $cc(A)$. Given a smooth symmetric function $f\colon[-1,1]^d\to\mathbb R$ with $f(0)>0$, I am interested in the ...
1 vote
1 answer
642 views

Interchange of integration order (of a not absolutely convergent integral with sinus)

Can we interchange the integral order of this integral to start integration on $x$ ? (Taking $g$ and $f$ two functions of rapid decrease which are $o(x^2)$ near zero) $$A=\int_{0}^\infty \int_0^{\...
1 vote
1 answer
262 views

Relationship between $f(t,x)$ as $t \to \infty$ and $f(t/\epsilon, x/\epsilon^2)$ as $\epsilon \to 0$ (periodic functions)

Let $f: (0,\infty)\times \mathbb {R} \to \mathbb{R}$ be $1$-periodic in the second variable and in $L^\infty((0,\infty)\times \mathbb{R}).$ If it is necessary, we can also assume $f$ to be continuous. ...
7 votes
1 answer
489 views

When the value of a function in a point is equal to its integral average over the point's neighborhood?

It is well-known that the harmonic functions have this remarkable Averaging Property: if $f$ is harmonic in a domain $U \subset R^n$, then, for any point $x \in U$, $f(x)$ is equal to the integral ...
5 votes
2 answers
840 views

Decompostition of a Lipschitz domain

We say that $\Omega$ is a strongly star shaped domain (with respect to $0$ for example) in $\mathbb R ^n$ if: $$\Omega = \{x\in \mathbb R ^n : \left \| x \right \| < g(\frac{x}{\left \| x\right \...
4 votes
1 answer
1k views

For what nonnegative measures $\mu$ does $\mu*e^{-|\cdot|}\in L^{\infty}$?

I am trying to characterize all measures on $\mathbb{R}$ such that $$ \sup_{x\in\mathbb{R}} \: (\mu*f)(x)<+\infty, $$ where $f(x)$ is some specific integrable functions, such as $f(x)=e^{-|x|}$, ...
7 votes
0 answers
187 views

distance distributions on a hypersphere?

Fix a real number $0\leq t\leq 1$ and an integer $n>1$. Let $\mathbb{S}^{n-1}\subset\mathbb{R}^n$ denote the unit hypersphere. Define $$d_N(n;t):=\max\sum_{i<j}\Vert P_i-P_j\Vert_2^t$$ where ...
0 votes
0 answers
58 views

in search of convergent daughter sequences

Let $\{f_n\}\subset L^1(\Omega,\mu)$, where $\mu$ is the Lebesgue measure, and $\Vert f_n\Vert_1\leq M$ and $\Vert Df_n\Vert_{1/2}\leq C$ uniformly in $n$. Question. Is there a subsequence $\{f_{...
3 votes
2 answers
435 views

A possible norm on a subspace of $C^\infty([0,1])$?

I have posted the following question (with minimal differences) on MSE some days ago, without receiving a satisfactory answer, so let me try here again. Take the vector space of infinitely ...
1 vote
1 answer
168 views

Does the Abel transform preserve analyticity?

Let $I=(0,1]$ and $T=\{(x,y)\in I^2;x\geq y\}$. If functions $f:I\to\mathbb R$ and $w:T\to\mathbb R$ are analytic, is the function $A_wf:I\to\mathbb R$, $$ A_wf(y)=\int_y^1\frac{f(x)w(x,y)}{\sqrt{x^2-...
7 votes
1 answer
609 views

$H^s$ norm of a solution of a nonlinear Schrödinger equation

I'm reading the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$ by Colliander, Keel, Staffilani, Takaoka and Tao. They study the ...
5 votes
0 answers
364 views

Version of Stone Weierstrass for functions not vanishing at infinity

I am trying to see what is known about uniform density of function spaces in $C(\mathbb{R}^n)$ or $C_b(\mathbb{R}^n)$ (bounded continuous functions on $\mathbb{R}^n$). By uniform density, I mean ...
7 votes
1 answer
306 views

An indicator of a planar subset as an element of a tensor product

Denote $I=(0, 1)$, and let $\mu$ be the Lebesgue measure on $I$. Does there exist a function $f$ on $I\times I$ viewed as an element of the space $L^\infty(\mu\times\mu)$ such that $$ f^2=f $$ (that ...
5 votes
1 answer
136 views

Reference for higher order Campanato Lemmas, e.g. `Sufficiently fast L^2 decay on balls to affine functions implies C^{1,\alpha}'

Whence can I reference the following fact (I have seen it quoted as `standard' in respectable places, so I hope it is so)?: Let $f : B_2(0) \to \mathbb{R}$, say $f \in L^2(B_2(0))$ . Suppose that ...
9 votes
3 answers
4k views

Is there a reference for compact imbedding theory of Hölder space?

This question is posted and unanswered from math.stackexchange. Suppose $0 < \alpha < \beta$ and $\Omega$ is bounded. Then, the Hölder space $C^\beta(\Omega)$ is compactly imbedded to $C^\alpha(...
1 vote
2 answers
931 views

A question on the Lebesgue differentiation theorem

In the paper [Jessen, B., Marcinkiewicz, J., and Zygmund, A. Note on the differentiability of multiple integrals. Fundamenta Mathematicae 25.1 (1935): 217-234] it is considered the limit $$ \lim_{\...
3 votes
1 answer
1k views

A calculus question related to the nonnegative definite functions

I am looking for some sufficient conditions for an even, continuous, nonnegative, non increasing function $f(x)$ on $R$ such that $$ \int_0^\infty \cos(xz) f(z) d z \ge 0 \qquad\text{for all $x\ge 0$...
0 votes
0 answers
153 views

extension of function in an abstract metric space

my question is the following.(Maybe my title is not quite proper for this question): Let $(E,d)$ be a Polish space (or a separable metric space), let $\xi: E\to R_+$ be a Lipschitz function. Now set $...
0 votes
1 answer
321 views

Is the span of those vectors dense in $\ell_2$?

For all $x \in \mathbb{R}^n$ and $\alpha \in \mathbb{Z}_{\geq 0}^n$ let $x^\alpha=x_1^{\alpha_1} \cdots x_n^{\alpha_n}$. Let $$\ell^2=\{z=(z_\alpha)_{\alpha \in \mathbb{Z}_{\geq 0}^n}:\, z_{\alpha} \...
4 votes
1 answer
370 views

Norms for complex measures

I'm searching for a state of the art introduction to norms on the space of complex measures (on $\mathbb R^n $, for example, or some compact subset thereof). I'd be interested in inequalities of the ...
5 votes
2 answers
774 views

Can we calculate the inner product of a semicontinous function with the Dirac delta function?

Dear all, It is clear that if $f:R\mapsto R$ is a continuous function, than $< f, \delta_x >=f(x)$. Now, if $f$ is only semicontinous, can we say that $< f, \delta_x >=f(x)$? I think this ...
4 votes
1 answer
471 views

Ask for theory about the weighted L^2(R^d) space.

Dear MOs, I am now considering the following norm: $$ ||f||_{H}^2 := \iint f(x) H(x,y) f(y) d x d y\:. $$ where the integral is over the whole space $R^{2d}$ and $H(x,y)$ is some non-negative ...
5 votes
1 answer
540 views

Cosets of groups of functions

Let's consider an interval $I\subseteq\mathbb R$, and let $\mathcal F(I)$ be the set of bijective functions $f:I\to I$ so that the graph of $f$ is a analytic curve in $I\times I$. The set $\mathcal ...
6 votes
2 answers
4k views

Is there dual space of the distributions $\mathcal{D}'(R)$?

Dear MOs, Let $\mathcal{D}(R):=C_c^\infty(R)$ be the smooth functions with compact support. Its dual space is the space $\mathcal{D}'(R)$ of distributions. This space $\mathcal{D}(R)$ has its weak *-...

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